Finite Group Databasebeta

← #29665293

Subgroups — #29665293

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456740C211yesmaximal, characteristic
2#475×D61C211yesmaximal, characteristic, complemented
3#21455286C211yesmaximal, characteristic
4#21454898C211yesmaximal, characteristic, complemented
5#21445250C211yesmaximal, characteristic, complemented
6#21445249C211yesmaximal, characteristic, complemented
7#21445248C211yesmaximal, characteristic, complemented
8#21445247C211yesmaximal, characteristic, complemented
9#21445166C211yesmaximal, characteristic, complemented
10#21445165C211yesmaximal, characteristic, complemented
11#21445164C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic
15C61×#1666C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17#16840310C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840134C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840048C2211yescharacteristic
24#16840047C2211yescharacteristic
25#16840046—21no—
26#16840045—21no—
27#16840044C2211yescharacteristic
28#16839983—21no—
29#16839982—21no—
30#16839973C2211yescharacteristic
31#16839972C2211yescharacteristic
32#16839965—21no—
33#16839965—21no—
34#16839959C2211yescharacteristic
35#16839899—21no—
36#16839899—21no—
37Q8×#92418C2211yescharacteristic
38C8×Dic122—21no—
39C8×Dic122C2211yescharacteristic
40#16839886—21no—
41#16839886C2211yescharacteristic
42#16839814—21no—
43#16839814C2211yescharacteristic
44#16839813—21no—
45#16839813C2211yescharacteristic
46#106×D61—21no—
47#16839784C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839760—21no—
50#103×D61—21no—
51#16839513C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839509C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839508C2211yescharacteristic
57#16839507C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839506C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#475C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×Q8×D61D411yescharacteristic
70#14519806—44no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758189—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519775—21no—
83#14519764—21no—
84#14519748—21no—
85#14519748—21no—
86Q8×Dic61D411yescharacteristic
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89#14519742—21no—
90#14519742D411yescharacteristic
91D4×Dic61—21no—
92D4×Dic61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61D411yescharacteristic
97C42×D61—21no—
98C42×D61C2311yescharacteristic
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519702—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519700—21no—
120#14519700—21no—
121#14519698—21no—
122#14519689—21no—
123#14519689D411yescharacteristic
124#14519689D411yescharacteristic
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C2×SD16×C61—21no—
132C2×SD16×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134#127×C61C2311yescharacteristic
135C4×Q8×C61C2311yescharacteristic
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×C22⋊D61—41no—
151C22×Dic61—41no—
152Q8×D61—22no—
153Q8×D61—81no—
154Q8×D61—81no—
155Q8×D61SD1614yescomplemented
156#11758224—42no—
157#11758224—42no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—41no—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61#4111yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168C2×Dic122C2×D411yescharacteristic
169SD16×C61—42no—
170SD16×C61—81no—
171SD16×C61—81no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C42×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185#11758202—42no—
186#11758200—42no—
187#11758200—81no—
188#11758200—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61#4111yescharacteristic
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211Dic244—42no—
212#11758189—42no—
213C8×D61—42no—
214C23×C61—41no—
215C22×D61#12111yescharacteristic
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61C2×SD1612yes—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—81no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C22⋊D61—82no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237C2×Dic61#12111yescharacteristic
238C2×Dic61#12111yescharacteristic
239C2×Dic61#12111yescharacteristic
240C4×D61—24no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—41no—
244C4×D61—81no—
245C4×D61—82no—
246C4×D61C2×SD1612yes—
247C4×D61#13512yes—
248Dic122—21no—
249Dic122—24no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—82no—
255Dic122C2×SD1612yes—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#19397—611nomaximal, Sylow
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12111yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61#46512yes—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61C22×SD1611yescharacteristic
279C4×C61C2×#13511yescharacteristic
280C4×C61#57711yescharacteristic
281C4×C61#57711yescharacteristic
282Dic61—21no—
283Dic61—22no—
284Dic61—21no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288Dic61#46512yes—
289#3206—611no—
290C2×#475—611no—
291#3037—611no—
292#2982—611no—
293#1680—611no—
294#1680—611no—
295#1680—611no—
296#1680—611no—
297#1666—611no—
298#1666—611no—
299#1666—611no—
300#1666D6111yescharacteristic, complemented
301#1570—611no—
302#1549—611no—
303#1540—611no—
304C2×C61—161no—
305C2×C61#304311yescharacteristic
306C2×C61C2×#46511yescharacteristic
307C2×C61#306911yescharacteristic
308D61—22no—
309C2×#137—1221no—
310#532—611no—
311C2×C4⋊Q8—611no—
312C2×C4×Q8—611no—
313#492—611no—
314#492C2×D6111yescharacteristic
315#487—611no—
316#487—611no—
317#475—611no—
318#475—611no—
319#475—611no—
320#475—1221no—
321#475—1221no—
322#475C2×D6111yescharacteristic
323#464—1221no—
324#464—1221no—
325#462—611no—
326#462—611no—
327#461—1221no—
328#461—1221no—
329#460—611no—
330#460C2×D6111yescharacteristic
331#448—1221no—
332#448—1221no—
333C8×Q8—611no—
334C8×Q8—611no—
335C8×Q8—611no—
336C8×Q8—1221no—
337C8×Q8—1221no—
338C8×Q8C2×D6111yescharacteristic
339#431—611no—
340#431—611no—
341#431—1221no—
342#431—1221no—
343C2×#106—1221no—
344#423—611no—
345C2×C4⋊C8—611no—
346#417—1221no—
347C2×#103—1221no—
348#334—611no—
349#334—611no—
350#334—611no—
351#334C2×D6111yescharacteristic
352#332—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356#332—611no—
357#332C2×D6111yescharacteristic
358#331—611no—
359#331C2×D6111yescharacteristic
360C61#1939711yescharacteristic, complemented, Sylow
361C2×#41—1221no—
362C22×Q8—611no—
363C22×Q8—1221no—
364#137—2444no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×M16—1221no—
374C22×C8—1221no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8C22×D6111yescharacteristic
381#127C22×D6111yescharacteristic
382#124—1221no—
383#121—1221no—
384#121—1221no—
385C4×Q8—611no—
386C4×Q8—611no—
387C4×Q8—611no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C22×D6111yescharacteristic
391C4×D4—1221no—
392C4×D4—1221no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C42—611no—
398C2×C42—1221no—
399#106—21no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403C4⋊C8—21no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415C4⋊C8C22×D6111yescharacteristic
416C4⋊C8C22×D6111yescharacteristic
417#103—21no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#102—21no—
428#102—1221no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C22⋊C8—1221no—
432C4×C8—21no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8—1221no—
437C4×C8—1221no—
438C4×C8C22×D6111yescharacteristic
439#41—2442no—
440#41—2442no—
441C2×Q8—21no—
442C2×Q8—614no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—1222no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—4881no—
449C2×Q8—4881no—
450C2×Q8—2441no—
451C2×Q8D4×D6111yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—2441no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—2441no—
462C22×C4—2441no—
463Q16—2442no—
464Q16—2442no—
465SD16—42no—
466SD16—81no—
467SD16—81no—
468SD16—4881no—
469SD16—4881no—
470SD16—2442no—
471M16—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2442no—
481C2×C8D4×D6111yescharacteristic
482C2×C8D4×D6111yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4D4×D6111yescharacteristic
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C42—21no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—1221no—
505C42—1221no—
506C42—1221no—
507C42C23×D6111yescharacteristic
508C23—41no—
509C23—611no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—612no—
514Q8—1224no—
515Q8—1221no—
516Q8—4882no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521Q8SD16×D6112yes—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—81no—
527D4—4882no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—1224no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—4881no—
542C2×C4—4882no—
543C2×C4—4882no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×D4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—1221no—
556C8—1221no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—612no—
564C22—1221no—
565C22—1221no—
566C22#121×D6111yescharacteristic
567C4—22no—
568C4—21no—
569C4—41no—
570C4—81no—
571C4—612no—
572C4—1222no—
573C4—1221no—
574C4—1221no—
575C4—4881no—
576C4—4881no—
577C4—2441no—
578C4C2×SD16×D6111yescharacteristic
579C4#135×D6111yescharacteristic
580C4#1684056811yescharacteristic
581C4#1684057211yescharacteristic
582C2—161no—
583C2—1222no—
584C2#2145533511yescharacteristic
585C2#465×D6111yescharacteristic
586C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1666C2
socleC22×C61C2×#121
prime radical#16840310C22
hypercenter#1666D61
hypocenterC61#19397
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1666

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #475×D61C2semidirect
4. #21454898C2semidirect
5. #21445250C2semidirect
6. #21445249C2semidirect
7. #21445248C2semidirect
8. #21445247C2semidirect
9. #21445166C2semidirect
10. #21445165C2semidirect
11. #21445164C2semidirect
12. #21444588C2semidirect
13. #21444502C2semidirect
15. C61×#1666C2semidirect
155. Q8×D61SD16semidirect
300. #1666D61semidirect
360. C61#19397semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214567402
#475×D612
#214552862
#214548982
#214452502
#214452492
#214452482
#214452472
#214451662
#214451652
#214451642
#214445882
#214445022
#214444652
C61×#16662
#1939761